Abstract
This article modifies the ratio test so that we can detect the change which happens in the latter half observations effectively. Moreover, we propose a new ratio test on the self-normalized numerator and the self-normalized denominator respectively. We establish the asymptotic properties under the null and alternative hypotheses. We also propose a new estimator to locate the early or later change. We check the theoretical validity of the block bootstrap approximation for the modified tests and give the weak convergence of the proposed estimator. Simulations demonstrate the validity of the block bootstrap procedures and the advantage of the proposed estimator. An application on a real data is illustrated for the practicability of the proposed test and the estimators.
Acknowledgment
The authors would like to thank the editor and the referees for their suggestions, which improved this manuscript greatly.